Theoretical Ecology (2020) 13:519–536 https://doi.org/10.1007/s12080-020-00465-8 ORIGINAL PAPER Threshold harvesting as a conservation or exploitation strategy in population management Frank M. Hilker1 · Eduardo Liz2 Received: 8 May 2019 / Accepted: 18 May 2020 / Published online: 20 June 2020 © The Author(s) 2020 Abstract Threshold harvesting removes the surplus of a population above a set threshold and takes no harvest below the threshold. This harvesting strategy is known to prevent overexploitation while obtaining higher yields than other harvesting strategies. However, the harvest taken can vary over time, including seasons of no harvest at all. While this is undesirable in fisheries or other exploitation activities, it can be an attractive feature of management strategies where removal interventions are costly and desirable only occasionally. In the presence of population fluctuations, the issue of variable harvests and population sizes becomes even more notorious. Here, we investigate the impact of threshold harvesting on the dynamics of both population size and harvests, especially in the presence of population cycles. We take into account semelparous and iteroparous life cycles, Allee effects, observation uncertainty, and demographic as well as environmental stochasticity, using generic mathematical models in discrete time. Our results show that threshold harvesting enhances multiple forms of population stability, namely persistence, constancy, resilience, and dynamic stability. We discuss plausible choices of threshold values, depending on whether the aim is resource exploitation, pest control, or the stabilization of fluctuations. Keywords Fixed escapement · Harvest control rule · Pest control · Maximum sustainable yield · Intervention frequency · Population cycles · Stability Introduction levels, detection thresholds, or values that can be used to justify spending a budget.1 The management of ecosystems often requires population The threshold is sometimes also called the escapement harvesting. Examples include the control of invasive species, (or biomass reference point in the fisheries literature), removal of weeds, roguing of infected plants, regulation because it corresponds to the population level that “escapes” of pests such as defoliating insects, and the harvesting of harvesting. As a harvesting strategy that keeps the pop- fisheries and forestry. The strategy of threshold harvesting ulation at a constant ceiling, threshold harvesting can be removes the entire excess of a population stock above a seen as the antipode to the constant-quota harvesting strat- certain threshold value and takes no harvest if the stock egy, which removes a constant quota from the population. is below the threshold. In many instances, this strategy Proportional harvesting, which removes a constant fraction emerges naturally, e.g., when pests exceed economic injury of the population, corresponds to an intermediate strat- egy between the two extremes of threshold harvesting and constant-quota harvesting. Figure 1 illustrates the three har- Frank M. Hilker vesting strategies by showing their relative harvest mortality [email protected] 1Equivalent effects on a population can be caused by mechanisms Eduardo Liz other than harvesting. The earliest examples we are aware of are given [email protected] by Ricker (1952) (and reiterated in his famous paper Ricker 1954, Fig. 34 and pp. 609–616). He describes the situation when a population is regulated by generalist predators in such a way that only individuals 1 Institute of Mathematics and Institute of Environmental in refuges of a given size escape predation or that predators switch their Systems Research, Osnabr¨uck University, 49069 Osnabr¨uck, search image abruptly at a certain prey threshold density. He refers to Germany two empirical examples, one being predation upon bob-white in Iowa 2 Departamento de Matematica´ Aplicada II, Universidade and the other one being predation upon Atlantic salmon parr in some de Vigo, 36310 Vigo, Spain rivers. 520 Theor Ecol (2020) 13:519–536 Threshold harvesting is known under many different names in quite different disciplines (see Table 1). Here, we use the term “threshold harvesting” because it seems the most inclusive one for the wide range of applications that we have in mind, stretching from pest control to resource exploitation. To avoid confusion, we point out that the term “threshold harvesting” is sometimes also used for other harvesting strategies, where there are different forms of removal above the threshold (e.g., Kaitala et al. 2003; Enberg 2005). According to a review of harvesting strategies, threshold harvesting generally performs best for maximizing cumu- lative yield, mean annual yield, and profit in the absence of observation error (Deroba and Bence 2008). This has been found to hold under a wide range of assumptions (e.g., Ricker 1958;Clark1976; Reed 1978, 1979, 1980; Ludwig 1979, 1980, 1998; Mendelssohn 1979; Mangel 1985; Lande et al. 1995, 1997). As harvesting is curtailed when the pop- ulation is below the threshold level, threshold harvesting Fig. 1 Relative harvest mortalities for threshold harvesting (blue, has a built-in mechanism to accommodate random varia- dotted), proportional harvesting (black, dashed), and constant-quota harvesting (red, solid). We denote by x the population size at tion in environmental conditions or vital rates (Lande et al. equilibrium, Y(x) the corresponding yield, T the threshold, c the 1997). This has two major consequences. On the one hand, constant quota, and q the proportional rate of harvest this makes threshold harvesting a precautionary strategy that prevents overexploitation and maintains spawning biomass. On the other hand, threshold harvesting may impose a “stop- (or per-capita harvest rate) as a function of population and-go fishery” (Evans 1981), which implies an economic abundance. For threshold harvesting, the relative population and political cost to more precautionary harvesting. Ricker removal ceases as the population becomes smaller and stops (1958) was perhaps the first to observe that the variability altogether when the population drops below the thresh- of catch is greater for threshold harvesting than for some old. By contrast, for constant-quota harvesting, the relative other strategies (see also Larkin and Ricker 1964;Tautz population removal increases as the population becomes et al. 1969; Allen 1973; Gatto and Rinaldi 1976). Another smaller. For proportional harvesting, the relative population drawback of threshold harvesting is that it requires up-to- removal is constant. date knowledge of stock size (Hilborn and Walters 1992). Table 1 A collection of the many names of threshold Name References harvesting Ecology, fisheries, natural resources management Threshold harvesting Lande et al. (1995) and Lande et al. (1997) Fixed escapement Hilborn (1979) and Fryxell et al. (2005) Constant escapement Reed (1979) and Getz and Haight (1989) Fixed stock Jonzen´ et al. (2002) Constant-stock-size strategy Hilborn and Walters (1992) Optimal strategy Ludwig (1998) Bang–bang Clark (1976) Optimum stabilization of escapement Ricker (1958) Trigger harvest Sabo (2005) (upper) limiter control Hilker and Westerhoff (2006) and Tung et al. (2014) Physics, telecommunication, nonlinear dynamics Thresholding Sinha (1994) and Sinha (2001) Flat-topped maps Glass and Zeng (1994) Simple limiter control Corron et al. (2000) and Stoop and Wagner (2003) Theor Ecol (2020) 13:519–536 521 Errors in stock size estimates may (Engen et al. 1997; account stochasticity in the form of demographic and auto- Milner-Gulland et al. 2001; Lillegard˚ et al. 2005)ormay correlated environmental variations as well as observation not (Butterworth and Bergh 1993; Polacheck et al. 1999) error, and investigates the harvest frequency and the vari- change the relative superiority of threshold harvesting com- ability of yield as a function of harvesting intensity. When pared to other strategies. In practice, thresholds are typically we say increased threshold harvesting we mean setting a set too low (Ludwig et al. 1993). Yet, it remains that thresh- lower threshold (escapement) which amounts to elevated old harvesting is a strategy minimizing extinction risk while harvesting intensity. In each of these sections, we not only optimizing yield (Lande et al. 1995; Lande et al. 1997) present results, but also place the models in the context and could thus bring together conservation and exploitation of the literature and discuss the relevance of our findings. interests. “Conclusions” provides a summary of the results and draws Recent microcosm experiments have investigated thresh- overall conclusions. old harvesting of populations of the ciliate Tetrahy- mena thermophila (Fryxell et al. 2005) and the fruit fly Drosophila melanogaster (Tung et al. 2016), for selected Population dynamic effects of threshold threshold values. Both studies support the utility of thresh- harvesting old harvesting as a management strategy. In both experi- ments, threshold harvesting was found to reduce variability In this Section, we study single-species populations without in population density and to reduce extinction risk. Allee effect managed by threshold harvesting. Let xn be Depending on the context, harvesting may be desirable the population size at time step n, n = 0, 1, 2,.... because it generates revenue (e.g., in fisheries) or unde- The population dynamics in the absence of harvesting is sirable because it causes expenditures